Global attractivity of positive periodic solution of a delayed Nicholson model

Global attractivity of positive periodic solution of a delayed Nicholson model

with nonlinear density-dependent mortality term

Doan Thai Son, Le Van Hien

and Trinh Tuan Anh

Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam Received 22 January 2018, appeared 3 February 2019

Communicated by Leonid Berezansky

Abstract. This paper is concerned with the existence, uniqueness and global attractivity of positive periodic solution of a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality rate. By some comparison techniques via differential in- equalities, we first establish sufficient conditions for the global uniform permanence and dissipativity of the model. We then utilize an extended version of the Lyapunov functional method to show the existence and global attractivity of a unique positive periodic solution of the underlying model. An application to the model with constant coefficients is also presented. Two numerical examples with simulations are given to illustrate the efficacy of the obtained results.

Keywords: Nicholson’s blowflies model, positive periodic solution, attractivity, time- varying delays, nonlinear density-dependent mortality.

2010 Mathematics Subject Classification: 34C25, 34K11, 34K25.

1 Introduction

Mathematical models are important for describing dynamics of phenomena in the real world [7,12,25]. For example, in [24], Nicholson used the following delay differential equation

N⁰(t) =−αN(t) +βN(t−τ)e^{−γN(t−τ)}, (1.1) where α, β, γ are positive constants, to model the laboratory population of the Australian sheep-blowfly. In the biology interpretation of equation (1.1), N(t) is the population size at time t, αis the per capita daily adult mortality rate, β is the maximum per capita daily egg production rate, _γ¹ is the size at which the population reproduces at its maximum rate and τ≥ 0 is the generation time (the time taken from birth to maturity). Model (1.1) is typically referred to the Nicholson’s blowflies equation. It is interesting that when the maximum re- producing rate is not limited (i.e. _γ¹ → +∞) and the time τ is small which can be ignored,

BCorresponding author. Email: hienlv@hnue.edu.vn

model (1.1) is reduced to a well-known model in population dynamics namely logistic growth model described as

N⁰(t) =−αN(t)

1− ^N(t) K

, (1.2)

whereK= ^α_β is a constant involving the environment capacity.

In the past few years, the qualitative theory for Nicholson model and its variants has been extensively studied and developed [1,2,9]. In particular, the problems associated with asymp- totic behavior of positive periodic and almost periodic solutions of Nicholson-type models with delays were studied in [15,19,20,29]. Nicholson-type models with stochastic pertur- bations and harvesting terms were also investigated in [30] and [6,17,23,27], respectively.

Very recently, in [3], the problems of stability and attractivity were studied for a class of n- dimensional Nicholson systems with constant coefficients and bounded time-varying delays.

Based on some comparison techniques in the theory of monotone dynamical systems, delay- dependent sufficient conditions were derived for the existence and global exponential stability of a unique positive equilibrium.

Most of the existing works so far are devoted to Nicholson-type models with linear mortal- ity terms. As discussed in [1], a model of linear density-dependent mortality rate will be most accurate for populations at low densities. According to marine ecologists, many models in fishery such as marine protected areas or models of B-cell chronic lymphocytic leukemia dy- namics are suitably described by Nicholson-type delay differential equations with nonlinear density-dependent mortality rate of the form [1]

N⁰(t) =−D(N(t)) +βN(t−τ)e^{−γN(t−τ)}, (1.3) where the functionD(N)might have one of the forms D(N) = a−be^−N (type-I) or D(N) =

aN

b+N (type-II) with positive constantsaandb. A natural extension of (1.3) to the case of variable coefficients and delays, which is more realistic in the theory of population dynamics [13,18], is given by

N⁰(t) =−D(t,N(t)) +β(t)N(t−τ(t))e^{−γ(t)N(t−τ(t))}, (1.4) where D(t,N) = a(t)−b(t)e^−N or D(t,N) = _b^a₍⁽_t^t₎₊^)N_N. In model (1.4), D(t,N) is the death rate of the population which depends on time t and the current population level N(t)_, B(t,N(t−τ(t))) = β(t)N(t−τ(t))e^{−γ(t)N(t−τ(t))} is the time-dependent birth function which involves a maturation delayτ(t)and gets its maximum _γ^β₍⁽_t^t₎⁾_e at rate _γ(¹_t). Recently, Nicholson- type models with nonlinear density-dependent mortality terms have attracted considerable research attention. In particular, some results on the permanence property of certain types of Nicholson models with delays were established in [16,21]. The existence and exponential stability of positive periodic solutions, almost periodic solutions of various Nicholson-type models with both type-I and type-II nonlinear mortality terms were considered in [4,5,18]

and [22,26,32,33], respectively. The author of [14] investigated the problem of global asymp- totic stability of zero-equilibrium of the following special model

N⁰(t) =−a(t) +a(t)e^−N(t)+

∑

β_j(t)N(t−τ_j(t))e^{−γj(t)N(t−τj(t))}. (1.5) It has been shown that under the restrictions

sup

t≥0

γ_j(t)≤1, sup

t≥0

∑

β_j(t)

a(_t)γ_j(_t) <1, (1.6)

the equilibrium N^∗ =0 of (1.5) is globally asymptotically stable with respect to an admissible phase space called C+. In recent work [31], the effect of delay on the stability and attractivity was studied for the following model

N⁰(t) =−a+be^−N(t)+

∑

β_jN(t−τ_j(t))e^{−γjN(t−τj(t))}, (1.7) where a,b, β_j andγ_j are constants. Under the restrictions

1 e

∑

β_j

γ_j < a, lnb

a > ¹

min₁≤j≤mγ_j, (1.8)

which ensure the existence of at least one positive equilibrium ¯N, it was shown by utilizing a technique called fluctuation lemma [25, Lemma A.1] that such an equilibrium ¯Nof (1.7) is globally attractive if the magnitude of delays satisfies the following condition

1max≤j≤msup

t≥0

τ_j(t)≤ ¹ aγ+¹_e∑^m_j=1β_j

, γ= max

1≤j≤mγ_j. (1.9) Clearly, condition (1.9) can only be applied for models with small delays. This will restrict the applicability of the obtained results to practical models.

Motivated by the above literature review, in this paper we study the problem of existence and global attractivity of positive periodic solution of the following Nicholson model

N⁰(t) =−D(t,N(t)) +

∑

β_k(t)N(t−τ_k(t))e^{−γk(t)N(t−τk(t))}, (1.10) where D(t,N) = a(t)−b(t)e^−N. Main contributions and innovation points of this paper are three folds. First, improved conditions on global uniform permanence and dissipativity of time-varying Nicholson models with nonlinear density-dependent mortality term in the form of (1.10) are derived based on new comparison techniques via differential inequalities. We do not impose restrictions on the maximum reproducing rates _γ¹

k like (1.6), (1.8) and (1.9).

Second, a novel approach to the problem of existence and global attractivity of a unique positive periodic solution of model (1.10) is presented. Third, as an application to Nicholson models with constant coefficients as (1.7), improved results on the existence, uniqueness and global attractivity of a positive equilibrium are obtained.

2 Preliminaries

For a given scalarω>0, a function f :R⁺→_Ris said to beω-periodic if f(t+ω) = f(t)for allt ≥0. LetPω(_R⁺)denote the set ofω-periodic functions onR⁺. Clearly, if f :R⁺ →_Ris a continuous function and f ∈ _Pω(_R⁺)for someω >0 then f is bounded onR⁺. Hereafter, for a bounded function f on [0,+_∞), we will denote

f⁺=sup

t≥0

f(t), f⁻=inf

t≥0f(t).

Consider a Nicholson model with delays and nonlinear density-dependent mortality term of the form

N⁰(t) =−D(t,N(t)) +

∑

β_k(t)N(t−τ_k(t))e^{−γk(t)N(t−τk(t))}, t ≥t₀, (2.1)

N(t) = ϕ(t), t∈[t₀−τ_M,t₀], (2.2)

where the density-dependent mortality termD(t,N)is of the form

D(t,N) =a(t)−b(t)e^−N (2.3) andτ_M = _max1≤k≤pτ_k⁺ represents the upper bound of delays. For more detail on biological explanations of the coefficients of system (2.1)–(2.3), we refer the reader to [1,3,31].

LetC , C([−τ_M, 0],R)be the Banach space of continuous functions on [−τ_M, 0]endowed with the norm kϕk = sup_{t∈[−τM,0]}|ϕ(t)| and C⁺ be the cone of nonnegative functions in C, that is,

C⁺={ϕ∈ C([−τ_M, 0],R): ϕ(t)≥0}.

We writeϕ≥0 forϕ∈ C⁺. In addition, a function ϕ∈ C is said to be positive, write asϕ>0, ifϕ(t)is positive for allt ∈[−τ_M, 0]. Due to the biological interpretation, the set of admissible initial conditions in (2.2) is taken as

C₀⁺=ϕ∈ C⁺ :ϕ(0)>0 .

Let us first introduce the following assumptions and conditions.

Assumption (A):

(A1) a,b,γ_k : [0,+_∞) → (0,+_∞), β_k : [0,+_∞) → [0,+_∞) and τ_k : [0,+_∞) → [0,τ_M] are continuous bounded functions, whereτ_M is some positive constant.

(A2) There exists anω>0 such that the functions a,b, β_k,γ_k andτ_k belong toPω(_R⁺). Condition (C):

(C1) a) b(t)≥ a(t)≥a⁻>0; b) θ,^{lim inf}t→+_∞^b(t) a(t) >1.

(C2) lim sup_{t→+∞ a(}¹_t)∑^p_k=1 βk(t)

γ_k(t) =σ; 1−^σ e >0.

(C3) b⁻−a⁺ >0; $, ^a−− ¹_e∑k^p₌₁ ^β+

γ⁻_k >0.

(C4) ∑^p_k=1β⁺_k maxn

1

e²,^{1−γ−kr∗}

e^γ−kr∗

o

< ^$b_b+⁻; r∗ =ln ^b_a⁻+

.

A preview of our main results is presented in the following table.

Conditions Results

(A1), (C1) Uniform permanence inC₀⁺, lim inf_t→+_∞N(t,t₀,ϕ)≥ln(θ). (A1), (C1a), (C2) Uniform dissipativity inC₀⁺, lim sup_t→+∞N(t,t₀,ϕ)≤ln _a−(^b1⁺−^σ_e)

. (A1), (C3) ln ^b_a⁻+

≤lim inft→+_∞N(t,t0,ϕ)≤lim sup_t→+∞N(t,t0,ϕ)≤ln ^b_$⁺ . (A1), (A2), (C3) There exists a unique positiveω-periodic solution N^∗(t)

and (C4) which is globally attractive inC₀⁺.

For a biological interpretation of the proposed conditions, it is reasonable that when the population is absence the death rate is nonpositive (i.e. D(t, 0) ≤ _{0) and} D(t,N) _{is always} positive when N > 0. This gives rise to condition (C1). On the other hand, in most of biological models, there typically exists a threshold related to the so-called carrying capacity.

When the population size is very large, over the carrying capacity, the death rate can be bigger than the maximum birth rate. Similarly to model (1.4), the quantity∑_k^p=1

β_k(t)

γk(t)e can be regarded

as the maximum birth rate of model (2.1). In addition, whenNis largeD(t,N)is approximate to a(t). By this observation, we make an assumption to ensure that ∑^p_k=1

βk(t)

γ_k(t)e < a(t). This reveals the imposing of condition (C2) when considering long-time behavior of the model.

(C3) is a testable condition derived from (C2) and (C1a) by taking into account the upper bound of the associated rates. While condition (C3) only guarantees non-extinction and non- blowup behavior, condition (C4) reveals that, by certain scaling coefficients, when maximum per capita daily egg production rates are smaller than the gap between the maximum death rate and birth rate (i.e. $ = a⁻− ¹_{e ∑k}^p₌₁ ^β+

γ⁻_k ), the population will be stable around a periodic trajectory (in the case of periodic coefficients) or a positive equilibrium (for time-invariant model).

In the remaining of this section, we present a local existence result of solutions of system (2.1)–(2.3). Fort >0, the function N_t ∈ C is defined as N_t(θ) = N(t+θ), θ ∈ [−τ_M, 0]. Then, system (2.1)–(2.2) can be rewritten in the following abstract form

N⁰(t) =F(t,Nt), t≥t0, Nt0 = ϕ, (2.4) where the function F:R⁺× C →_Ris defined by

F(t,ϕ) =−D(t,ϕ(0)) +

∑

β_k(t)ϕ(−τ_k(t))e^{−γk(t)ϕ(−τk(t))}. (2.5) Proposition 2.1. Under assumption (A1), for any t₀ ≥ 0, ϕ ∈ C, there exists a unique solution N(t,t0,ϕ)of system(2.1)–(2.2)defined on a maximal interval[t0,η(ϕ)).

Proof. Clearly, the function F(·,·) defined in (2.5) is continuous and locally Lipschitz with respect to ϕ. Thus, the existence and uniqueness of a local solution of (2.1)–(2.2) is straight- forward [8] and the proof is omitted here.

3 Permanence of global positive solutions

3.1 Global existence of positive solutions

Theorem 3.1. Let assumption (A1) hold. Assume that b(t) ≥ a(t) for all t ∈ [_0,+_∞). Then, for any initial condition ϕ∈ C₀⁺, the solution N(t,t₀,ϕ)of system(2.1)–(2.3)is positive, N(t,t₀,ϕ)>0, t∈[t₀,η(ϕ)), andη(ϕ) = +_∞.

The following lemma will be used in the proof of Theorem3.1.

Lemma 3.2. Let a(t), b(t) ≥ 0, t ∈ [0,+_∞), be given continuous functions, the unique solution of the initial value problem (IVP)

x⁰ =−a(t) +b(t)e^−x, t≥t₀≥0, x(t₀) =x₀, (3.1) is given by

x(t,t₀,x₀) =−

Z _t

t0

a(s)ds+_ln

e^x0+

Z _t

t0

e

Rs t0a(θ)dθ

b(s)ds

. (3.2)

Proof. Define ˆx=e^x then (3.1) is written as ˆ

x⁰ = −a(t)xˆ+b(t)_, t≥t₀, ˆx(t₀) =e^x0. (3.3)

Observe that (3.3) is an IVP of linear differential equations. Thus, from (3.3) we have ˆ

x(t) =e⁻

Rt

t0a(s)ds ˆ x(t₀) +

Z _t

t0

e

Rs t0a(θ)dθ

b(s)ds

>0, ∀t≥t₀,

which leads to (3.2). The proof is completed.

Proof. (of Theorem3.1) Clearly, N(t₀,t₀,ϕ) = ϕ(0)>0. Thus, for sufficiently small t−t₀ >0, N(t,t₀,ϕ) > 0. We first show that N(t,t₀,ϕ) is positive on [t₀,η(ϕ)). If this does not hold, then there exists a t₁ > t0 such thatN(t₁,t0,ϕ) = 0 and N(t,t0,ϕ) > 0 for allt ∈ [t0,t₁). In regard to initial condition ϕ ∈ C₀⁺, we have N(t,t₀,ϕ) ≥ 0, t ∈ [t₀−τ_M,t₁]. Thus, it follows from (2.1) that

N⁰(t)≥ −D(t,N(t)), t∈ [t0,t₁). (3.4) By the comparison principle, and by utilizing Lemma3.2, from (3.4) we obtain

N(_t,t0,ϕ)≥ −

Z _t

t0

a(_s)_ds+_ln

e^ϕ(0)+

Z _t

t0

e

Rs t0a(θ)dθ

b(_s)_ds

≥ −

Z _t

t0

a(s)ds+ln

e^ϕ(0)−1+e

Rt

t0a(s)ds

. (3.5)

Lett↑t₁, estimation (3.5) implies that 0=N(t₁,t₀,ϕ)≥ln

1+ e^ϕ(0)−1 e⁻

Rt1 t0 a(s)ds

>0,

which is obviously a contradiction. Therefore, N(t,t0,ϕ)>0 for allt∈ [t0,η(ϕ)). Next, by using the fact sup_x≥0xe^−γx = _γe¹ for anyγ>0, it follows from (2.1) that

N⁰(t)≤ b(t)e^−N(t)−a(t) +¹ e

∑

β_k(t)

γ_k(t)^{, t} ≥t0. (3.6) Ifη(ϕ) < +_∞then lim_t↑η(ϕ)N(t,t₀,ϕ) = +_∞[8, Theorem 3.2]. On the other hand, similarly to (3.2), from (3.6) we have

N(t,t₀,ϕ)≤ −

Z _t

t0

ψ(s)ds+ln

e^ϕ(0)+

Z _t

t0

e

R_s

t0ψ(θ)dθ

b(s)ds

, t ∈[t₀,η(ϕ)), (3.7) where

ψ(t) =a(t)− ¹ e

∑

β_k(t) γ_k(t)^.

Thus, lim_t↑η(ϕ)N(t,t₀,ϕ) < +_∞as η(ϕ) < +∞. This contradiction shows that η(ϕ) = +_∞.

The proof is completed.

Remark 3.3. In general, the estimation (3.7) does not guarantee the boundedness ofN(t,t₀,ϕ) on interval [0,+_∞). In order to get the boundedness of N(t,t₀,ϕ), additional conditions should be imposed. For example, if there exists a positive constantM such that

sup

t≥0

n

b(t)e^−M−ψ(t)^o<0,

where ψ(t)is defined as in (3.7), then N(t,t₀,ϕ) < M, t ∈ [t_ϕ,+_∞), for some t_ϕ > 0 [5,18].

For other types of conditions, we refer the reader to Theorem3.7in this paper.

Remark 3.4. To ensure the positiveness of solutions of (2.1)–(2.3) with initial conditions inC₀⁺, conditionb(t)≥a(t)cannot be relaxed. For a counterexample, letn=1 and assume that

sup

t≥0

b(t)

a(t) =δ∈ [0, 1),

Z _t

0 a(s)ds→+_∞, t→+_∞.

Then, it follows from (3.2) that N(t,t₀,ϕ)≤ln

e^ϕ(0)−

Rt t0a(s)ds

+δ

1−e⁻

Rt

t0a(s)ds

→ln(δ)<0 ast→+_∞.

3.2 Uniform permanence

Theorem 3.5. Let assumption (A1) hold. Assume that b(t)≥a(t)≥ a⁻>0and lim inf

t→+_∞

b(t)

a(t) ≥e^`m >1. (3.8)

Then, for anyϕ∈ C₀⁺,

lim inf

t→+_∞ N(t,t₀,ϕ)≥`_m >0.

Proof. By Theorem3.1, N(t,t₀,ϕ)>_{0 for all}t∈ [t₀,+_∞). On the other hand, for a sufficiently smalle>0, by (3.8), there exists aT>t₀such thatb(t)≥ (e^`m−e)a(t)for allt≥T. Similarly to (3.5), we have

N(t,t₀,ϕ)≥ln

e^{N(T,t0,ϕ)−RTta(s)ds}+ (e^`m−e)1−e^−RTta(s)ds . Note also that

Z _t

T a(_s)_ds→+_{∞ ast} →+_∞.

Thus, lett →+_∞ande↓0, we obtain lim inf

t→+_∞ N(t,t₀,ϕ)≥`_m. The proof is completed.

Remark 3.6. As a special case of (3.8), for bounded functions a(t) andb(t), if b⁻ > a⁺ then the scalar `_m in (3.8) can be chosen as

`_m =_ln

b⁻ a⁺

.

Thus, Theorem3.5in this paper encompasses the result of Lemma 1 in [31].

The following result shows the uniform dissipativity of system (2.1)–(2.3) inC₀⁺[3] in the sense that there exists a constant`<sub>M</sub> >0 such that lim sup<sub>t→+∞</sub>N(t,t<sub>0</sub>,ϕ)≤`_M.

Theorem 3.7. Assume assumption (A1) and the following conditions hold

b⁺≥b(t)≥a(t)≥ a⁻ >0, t∈[0,+_∞), (3.9) lim sup

t→+_∞

1 a(t)

∑

β_k(t)

γ_k(t) =σ, 1− ^σ

e >0. (3.10)

Then, system (2.1)–(2.3) is uniformly dissipative in C₀⁺. More precisely, for any initial condition ϕ∈ C₀⁺, the corresponding solution N(t,t0,ϕ)of (2.1)–(2.3)satisfies

lim sup

t→+_∞

N(t,t₀,ϕ)≤`_M ,^ln

b⁺ a⁻ 1− ^σ_e

.

Proof. By similar lines used in the proof of Theorem3.1, from (3.7), we have N(t,t₀,ϕ)≤ −

Z _t

t0

ψ(s)ds+ln

e^ϕ(0)+b⁺ Z _t

t0

e

Rs t0ψ(θ)dθ

ds

=ln

e^ϕ(0)e⁻

R_t

t0ψ(s)ds

+b⁺e⁻

R_t

t0ψ(s)dsZ _t

t0

e

R_s

t0ψ(θ)dθ

ds

, t∈ [t₀,+_∞). (3.11) On the other hand, by (3.10), there exists aT>0 such that

1−¹ e

∑

β_k(t)

a(t)γ_k(t) ≥1− ^σ

e >0, t≥ T.

Therefore,

ψ(t) =a(t)

1− ¹ e

∑

β_k(t) a(t)γ_k(t)

≥ a⁻ 1−^σ

e

>0, t ≥T.

By this,Rt

t0ψ(s)ds→+_∞ast→+∞, and thus lim sup

t→+_∞

e⁻

Rt

t0ψ(s)dsZ _t

t0

e

Rs t0ψ(θ)dθ

ds=lim sup

t→_∞

1

ψ(t) ≤ ¹ a⁻ 1− ^σ_e^.

Lett→+∞, from (3.11) we obtain lim sup_t→+∞N(t,t₀,ϕ)≤ `_M. The proof is completed.

The following result is obtained as a consequence of Theorems3.5and3.7.

Corollary 3.8. Let assumption (A1) hold, where a,b,β_k and γ_k are bounded functions, γ_k⁻ > 0.

Assume that

$,^a−−¹ e

∑

β⁺_k γ⁻_k

>0, (3.12a)

b⁻−a⁺ >0. (3.12b)

Then, for anyϕ∈ C₀⁺, it holds that

ln b⁻

a⁺

≤lim inf

t→+_∞ N(t,t₀,ϕ)≤lim sup

t→+_∞

N(t,t₀,ϕ)≤ln b⁺

$

. (3.13)

Remark 3.9. For bounded coefficientsa,b,β_k,γ_k, it follows from (2.5) that the function F(t,ϕ) maps any bounded set B ⊂ C into a bounded setF(t,B)in R. Thus, by the assumptions of Corollary3.8, for any ϕ∈ C₀⁺, F(t,N_t)is bounded. Consequently, the corresponding solution N(t,t₀,ϕ)is uniformly continuous on[_0,+_∞)_.

4 Global attractivity of positive periodic solution

The following lemmas will be used in the proof of our results in this section.

Lemma 4.1([10]). Let u:R⁺→_Rⁿbe a uniformly continuous function. If the limitlimt→+_∞Rt 0u(_s)_ds exists and is finite, thenlimt→+_∞ku(t)k=0.

Proof. A detailed proof was presented in [10]. Let us omit it here.

Lemma 4.2([10]). Let u: R⁺ →_Rⁿ be a bounded uniformly continuous function. If there exists an ω>0such that

Z _+∞

0

ku(t+ω)−u(t)kdt< +_∞,

then there exists a continuousω-periodic function u^∗(t)satisfyinglim_t→+_∞ku(t)−u^∗(t)k=0.

This lemma was stated in [10]. To make it easier to follow, in this paper, we will also reconduct the following proof.

Proof. Sinceuis bounded, there exists a constantu_∞ >0 satisfying ku(t)k ≤u_∞ for allt ≥0.

Besides that for a given e > 0 there exists a δ = δ(e) > 0 such that ku(t₁)−u(t₂)k < e whenever |t₁−t₂| < δ due to the uniform continuity of u. We now defined a sequence of functions u_k : R⁺ → _Rⁿ, k ∈ _N0, by u_k(t) = u(t+kω). Then, we have ku_k(t)k = ku(t+kω)k ≤ u_∞ for all t ≥ 0, k ∈ _N0, which shows the uniform boundedness of the sequence {u_k}. On the other hand,

ku_k(t₁)−u_k(t2)k=ku(t₁+kω)−u(t2+kω)k<e, ∀k∈_N0,

whenever|t₁−t₂|< δ. Thus, the sequence {u_k}is uniformly equicontinuous. By the Arzelà–

Ascoli Theorem, there exists a subsequence {u_kp} that converges uniformly on the interval [_0,ω]to a continuous function denoted as u^∗(t). Hence, for a givene>_0,

ku(t+k_pω)−u^∗(t)k< ^e

2ω, ∀t∈ [_0,ω]_, p≥ p_e, for some p_e ∈N, which yields

Z _ω

0

ku(t+kpeω)−u^∗(t)kdt< ^e 2. On the other hand, it follows from the assumption that

∑

Z _ω

0

ku(t+ (n+1)ω)−u(t+nω)kdt=

Z _∞

0

ku(t+ω)−u(t)kdt<_∞.

This ensures∑^∞_n=k

Rω

0 ku_n(t+ω)−u_n(t)kdt→0 ask →∞. Thus, we can assume without lost of generality that

∑

Z _ω

0

ku_n(t+ω)−u_n(t)kdt< ^e 2. Now, for any k>kp_e, we have

Z _ω

0

ku(t+kω)−u^∗(t)kdt≤

Z _ω

0

ku(t+kω)−u(t+kpeω)kdt+

Z _ω

0

ku(t+kpeω)−u^∗(t)kdt

≤

k−1 n=

∑

Z _ω

0

kun(t+ω)−un(t)kdt+ ^e 2

<e.

This shows thatRω

0 ku(t+kω)−u^∗(t)kdt → 0 as k → ∞. Then, it can be verified by similar arguments used in [10] thatu_k ⇒ ^{u∗ on} [0,ω]as k → ∞. It is clear that for any t ≥ 0 there exist a uniquek ∈ _N0 ands ∈ [0,ω)such that t = kω+s. We now span the function u^∗ by definingu^∗(t) = u^∗(s) then u^∗ is a continuous and ω-periodic function on [0,∞) satisfying ku(t)−u^∗(t)k= ku_k(s)−u^∗(s)k →0 ast =kω+s →∞. The proof is completed.

Remark 4.3. For positive scalarsθ₁≤θ₂andγ, we have max

θ1≤x≤θ2

|1−γx|e^−γx ≤max 1

e²,(1−γθ₁)e^−γθ1

.

In our latter derivations,γis typically time-varying with a lower bound namely γ≥γ⁻> 0.

Then, for a givenθ₁ > 0, max₁

e²,(1−γθ₁)e^−γθ1 = ¹

e² whenever γθ₁ ≥ 1. In addition, the function(1−γθ₁)e^−γθ1 is decreasing inγ∈ 0,_θ¹

1

. Thus, the estimation

|1−γx|e^−γx ≤max 1

e²,1−γ⁻θ₁ e^γ−θ1

(4.1) holds for allx∈ [θ₁,θ2]andγ≥γ⁻ >0.

Let f be a differentiable function. Similarly to [28], we define a generalized sign-function σ_f as follows

σ_f(t) =











1 if[f(t)>0]∨[f(t) =0∧ f⁰(t)>0], 0 if[f(t) =0∧ f⁰(t) =0],

−1 if[f(t)<0]∨[f(t) =0∧ f⁰(t)<0].

Then, it is clear that |f(t)| = f(t)σ_f(t). Moreover, by similar lines used in the proof of Lemma 3.1 in [28], we have the following lemma.

Lemma 4.4. For a differentiable function f , it holds that

D⁺|f(t)|,^{lim sup}

h→0⁺

|f(t+h)| − |f(t)|

h = f⁰(t)σ_f(t), where D⁺is the upper-right Dini derivative.

In the following, we assume that assumptions (A1), (A2) and conditions (3.12a)–(3.12b) are satisfied. For convenience, we denote

r∗=_ln b⁻

a⁺

, r^∗ =_ln b⁺

$

, ν_k =_max (1

e²,1−γ⁻_k r∗

e^γ−kr∗ )

.

Note that, by (3.12b), ^b_a⁻+ > 1 and hence r∗ > 0. In addition, since the condition r∗ < ¹

maxγ⁻_k

is not imposed, 1−γ⁻_k r∗ can be positive, negative or zero. For 1 ≤ k ≤ p that 1−γ_k⁻r∗ ≤ 0, ν_k = ¹

e².

We are now in a position to present the existence, uniqueness and global attractivity of a positive periodic solution of system (2.1)–(2.3) as in the following theorem.

Theorem 4.5. Let assumptions (A1), (A2), conditions (3.12a), (3.12b) and the following ones are satisfied

inft≥0

1−τ_k⁰(t) =µ>0, (4.2)

∑

ν_kβ⁺_k <µ$b⁻

b⁺ , (4.3)

where $is the constant defined in (3.12a). Then, system(2.1)–(2.3) has a unique positiveω-periodic solution N^∗(t)which is globally attractive inC₀⁺.

Proof. We divide the proof of Theorem4.5into the following three steps.

Step 1. Since condition (4.3) can be written as∑_k^p₌₁ν_kβ⁺_k −µe^−r∗b⁻<0, for a givene>0 such thatr∗−e>0, we have

∑

ν_k^eβ⁺_k −µe^−(r∗+e)b⁻<0, where

ν_k^e=max (1

e²,1−γ⁻_k (r∗−e) e^{γ−k(r∗−e)}

) .

Let N(_t) = _N(_t,t0,ϕ) be a solution of system (2.1)–(2.3) with initial condition ϕ ∈ C₀⁺. Then, by Corollary 3.8, there exists aT>t₀ such that

r∗−e≤ N(t)≤r^∗+_e, ∀t≥ T−τ_M.

We define the function N_ω(t) = N(t+ω)−N(t)and consider the following Lyapunov-like functional

V(t) =|N_ω(t)|

| {z }

V₁(t)

+

∑

ν_k^eβ⁺_k µ

Z _t

t−τ_k(t)

|N_ω(s)|ds. (4.4)

By virtue of the periodicity of τ_k, γ_k and other coefficients, and by utilizing Lemma4.4, the upper-right Dini derivative ofV₁(t)is computed and estimated as follows

D⁺V₁(t) =σ_Nω(t)(N⁰(t+ω)−N⁰(t))

=σ_Nω(t)

b(t)e^−N(t+ω)−e^−N(t)

+

∑

β_k(t)^hN(t+ω−τ_k(t+ω))e^{−γk(t+ω)N(t+ω−τk(t+ω))}

−N(t−τ_k(t))e^{−γk(t)N(t−τk(t))}i

≤σ_Nω(t)b(t)e^−N(t+ω)−e^−N(t) +

∑

β_k(t)

N(t+ω−τ_k(t))e^{−γk(t)N(t+ω−τk(t))}−N(t−τ_k(t))e^{−γk(t)N(t−τk(t))} . (4.5) By mean-value theorem,

b(t)σ_Nω(t)e^−N(t+ω)−e^−N(t)

=−b(t)σ_Nω(t)N_ω(t)e^−ζ(t),

whereζ(t)is some value betweenN(t)and N(t+ω). Therefore,

−b(t)σ_Nω(t)e^−N(t+ω)−e^−N(t)

≤ −b⁻e^−(r∗+e)|Nω(t)|, ∀t≥ T. (4.6) In regard to (4.1), we also have

xe^−γk(t)x−ye^−γk(t)y

≤ν_k^e|x−y|, t≥0, x,y∈[r∗−e,r^∗+e]. The above estimation gives

N(t+ω−τ_k(t))e^{−γk(t)N(t+ω−τk(t))}−N(t−τ_k(t))e^{−γk(t)N(t−τk(t))}

≤ν_k^e|N_ω(t−τ_k(t))|. (4.7) Combining (4.5)–(4.7) we then obtain

D⁺V₁(t)≤ −b⁻e^−(r∗+e)|N_ω(t)|+

∑

β⁺_kν_k^e|N_ω(t−τ_k(t))|. (4.8) Similarly to (4.5), for

V₂(t) =

∑

ν_k^eβ⁺_k µ

Z _t

t−τ_k(t)

|N_ω(s)|ds, we have

D⁺V₂(t) =

∑

ν_k^eβ⁺_k

µ |Nω(t)| −(1−τ_k⁰(t))|Nω(t−τ_k(t))|

≤

∑

ν^e_kβ⁺_k 1

µ|N_ω(t)| − |N_ω(t−τ_k(t))|. (4.9) It follows from (4.8) and (4.9) that

D⁺V(t)≤

" _p

k

∑

ν_k^eβ⁺_k

µ −b⁻e^−(r∗+e)

| {z }

−ρ

#

|Nω(t)|. (4.10)

Sinceρ>0, it follows from (4.10) that Z _+∞

T

|N_ω(t)|dt≤ ^V(T)

ρ <+_∞.

Note also that|N⁰(t)|is bounded since the right-hand side of (2.1) is bounded. Consequently, N(t)is a uniformly continuous function. By Lemma 4.2, there exists an ω-periodic function N^∗(t)such that|N(t)−N^∗(t)| →0 ast→+_∞.

Step 2.It follows from (2.4) that

N(t) = N(0) +

Z _t

0 F(λ,N_λ)dλ.

Thus, for anyn∈_N,

N(t+nω) = N(₀) +

Z _t+nω

0

F(_λ,N_λ)dλ

= N(nω) +

Z _t+nω

nω F(λ,N_λ)dλ

= N(nω) +

Z _t

0 F(λ,N_λ+nω)dλ (4.11)

due to the periodicity of the functions a,b,β_k andγ_k. Since N^∗(t) is anω-periodic function, whenntends to infinity, we have

N(nω) = N(nω)−N^∗(nω) +N^∗(₀)→ N^∗(₀)_, N_λ+nω = N_λ+nω−N_λ^∗_+nω+N_λ^∗ → N_λ^∗.

Letn→+_∞in (4.11) we obtain

N^∗(t) =N^∗(0) +

Z _t

0 F(λ,N_λ^∗)dλ, t ∈[0,ω].

This means that N^∗(t)is an ω-periodic solution of (2.1), which is also a positive solution of (2.1) according to Corollary3.8.

Step 3. We now prove that such an ω-periodic solution N^∗(t) of (2.1) is unique. Assume in contrary that ˆN^∗(t)is also anω-periodic solution of (2.1). Consider the following functional

W(t) =|N^∗(t)−N^ˆ∗(t)|+

∑

ν^e_kβ⁺_k µ

Z _t

t−τ_k(t)

|N^∗(s)−N^ˆ∗(s)|ds. (4.12) Similarly to (4.10), we have

D⁺W(t)≤ −b⁻e^−(r+e)|N^∗(t)−N^ˆ∗(t)|+

∑

ν_k^eβ⁺_k

µ |N^∗(t)−N^ˆ∗(t)|

≤ −ρ|N^∗(t)−N^ˆ∗(t)|, t≥ T.

This leads to

Z +_∞ 0

|N^ˆ∗(t)−N^∗(t)|dt<+_∞.

Since N^∗(t) and ˆN^∗(t) are uniformly continuous, which is deduced from Corollary 3.8, by Lemma4.1, limt→+_∞|N^ˆ∗(t)−N^∗(t)|= 0. Thus, for any δ >0 there exists aT∗ >0 such that

|N^ˆ∗(t)−N^∗(t)|<δ for allt≥ T∗. For anyt ≥0, withn∈_Nsatisfyingt+nω >T∗, we have

|N^ˆ∗(t)−N^∗(t)|=|N^ˆ∗(t+nω)−N^∗(t+nω)|<δ. (4.13) Letδ ↓0 in (4.13), we obtain ˆN^∗(t) = N^∗(t).

Finally, for any solution N(t,t₀,ϕ) of (2.1)–(2.3), it can be deduced from the arguments used in Steps 1 and 3 that |N(t,t₀,ϕ)−N^∗(t)| → _{0 as} t → +∞. This shows the global attractivity of N^∗(t). The proof is completed.

Remark 4.6. Conditions (4.2) and (4.3) are involved a scalar µ > 0 related to the rate of change of delay functionsτ_k(t). However, this scalar can be relaxed and conditions (4.2), (4.3) are reduced to the following one

∑

ν_kβ⁺_k < ^$b

−

b⁺ . (4.14)

More precisely, we state that in the following theorem.

Theorem 4.7. Under assumptions (A1) and (A2), assume that conditions(3.12a),(3.12b)and(4.14) are satisfied. Then, system(2.1)–(2.3)has a unique positiveω-periodic solution N^∗(t)which is globally attractive inC₀⁺.